Acoustical Experiments. 449 



modification of the sense of pitch is to be expected from 

 increased loudness. If the basilar membrane may be regarded 

 as being stretched with a finite tension in the direction of its 

 breadth, and as having no appreciable tension in the perpen- 

 dicular direction, then Helmholtz shows that it will be 

 vibrationally equivalent to a series of strings stretched side 

 by side and unconnected with one another. For shortness, I 

 shall speak of the membrane as if it actually consisted of such 

 separate strings, and thus, following HelmhohVs theory, we 

 are to suppose that a disturbance of given period reaching 

 the ear excites the strongest resonance in those strings whose 

 natural period is most nearly the same. Now when a string- 

 is vibrating freely with finite amplitude, the period of its 

 vibrations is shorter than if the amplitude were infinitesimal ; 

 and we are accordingly led to enquire whether a periodic 

 force of considerable intensity would not excite the maximum 

 resonance in those strings whose natural period for some finite 

 vibration-amplitude was most nearly the same. 



5. As sufficiently representative of our case we may take a 

 system with one degree of freedom, in which the positional 

 force contains a term proportional to the cube of the displace- 

 ment from equilibrium, as well as one proportional to the 

 first power*, the equation of motion being accordingly 

 written 



/ x 2 \ 

 l? = mx + kx + hxl 1+ — 2 Y . . . . (1) 



where F is the external force impressed on the system, x is 

 the displacement from the equilibrium position, and m, k, h, a 

 are real constants. 



If we take F to be a simple harmonic function of the time 

 whose character has been maintained long enough for the 

 whole motion to have become periodic, the value of x will be 

 expansible in a Fourier's series in which the constant coeffi- 

 cients have to be determined. But the expressions thus ob- 

 tained are very unwieldy; and it will therefore be more con- 

 venient to treat x instead of F as a simple harmonic function 

 of the time. It is not easy to say definitely which assump- 

 tion corresponds most nearly with the actual case, and from 

 what follows I think it will be evident enough that whichever 

 case we consider the general conclusions would be much the 

 same. 



* Terms of even degree would imply that the free vibrations were not 

 symmetrical with respect to the equilibrium position, and would there- 

 fore be absent in the case of a stretched string, which we have supposed 

 to agree pretty nearly with the physiological case. 



